A112031 -id:A112031 - cp's OEIS frontend

A112030 a(n) = (2 + (-1)^n) * (-1)^floor(n/2).

3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1, -3, -1, 3, 1

Offset: 0

Reinhard Zumkeller, Aug 27 2005

The fractions A112031(n)/A112032(n) give the partial sums of a(n)/floor((n+4)/2).

Sum of the two Cartesian coordinates from the image of the point (2,1) after n 90-degree counterclockwise rotations about the origin. - Wesley Ivan Hurt, Jul 06 2013

Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A010684, A016116, A112031, A112032, A112033.

A112030 := proc(n)
        (2 + (-1)^n) * (-1)^floor(n/2) ;
end proc: # R. J. Mathar, Jul 09 2013

LinearRecurrence[{0, -1}, {3, 1}, 100] (* Jean-François Alcover, Nov 24 2020 *)

a(n)=[3,1,-3,-1][n%4+1] \\ Charles R Greathouse IV, Aug 21 2011

def A112030(n): return (3, 1, -3, -1)[n&3] # Chai Wah Wu, Jan 31 2023

a(n) = A010684(n+1) * (-1)^floor(n/2).

O.g.f.: (3+x)/(1+x^2). - R. J. Mathar, Jan 09 2008

A112032 Denominator of 3/4 + 1/4 - 3/8 - 1/8 + 3/16 + 1/16 - 3/32 - 1/32 + 3/64 ...

Original entry on oeis.org

4, 1, 8, 2, 16, 4, 32, 8, 64, 16, 128, 32, 256, 64, 512, 128, 1024, 256, 2048, 512, 4096, 1024, 8192, 2048, 16384, 4096, 32768, 8192, 65536, 16384, 131072, 32768, 262144, 65536, 524288, 131072, 1048576, 262144, 2097152, 524288, 4194304, 1048576

Offset: 0

Reinhard Zumkeller, Aug 27 2005

Denominator of partial sums of A112030(n)/A016116(n+4), numerators = A112031;
A112031(n)/a(n) - 2/3 = (-1)^floor(n/2) / A112033(n);
lim_{n->infinity} A112031(n)/a(n) = 2/3.

G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 4, Sect. 1, Problem 148.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A016116, A112030, A112031, A112033.

[2^(Floor(n/2) + 1 + (-1)^n): n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

LinearRecurrence[{0,2},{4,1},50] (* following conjecture in Formula field above *) (* Harvey P. Dale, Dec 21 2014 *)

m=50; v=concat([4,1], vector(m-2)); for(n=3, m, v[n]=2*v[n-2]); v \\ G. C. Greubel, Nov 08 2018

a(n) = 2^(floor(n/2) + 1 + (-1)^n) = 2^A084964(n).
Conjectures from Colin Barker, Apr 05 2013: (Start)
a(n) = 2*a(n-2).
G.f.: (x+4) / (1-2*x^2). (End)

a(21) corrected by Vincenzo Librandi, Aug 17 2011

A112033 a(n) = 3 * 2^(floor(n/2) + 1 + (-1)^n).

Original entry on oeis.org

12, 3, 24, 6, 48, 12, 96, 24, 192, 48, 384, 96, 768, 192, 1536, 384, 3072, 768, 6144, 1536, 12288, 3072, 24576, 6144, 49152, 12288, 98304, 24576, 196608, 49152, 393216, 98304, 786432, 196608, 1572864, 393216, 3145728, 786432, 6291456, 1572864

Offset: 0

Reinhard Zumkeller, Aug 27 2005

George Pólya and Gábor Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chapter 4, Sect. 1, Problem 148.

Reinhard Zumkeller, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A084964, A112030, A112031, A112032.

A112033:=n->3*2^(floor(n/2) + 1 + (-1)^n); seq(A112033(k), k=0..50); # Wesley Ivan Hurt, Nov 01 2013

Table[3*2^(Floor[n/2] + 1 + (-1)^n), {n,0,50}] (* Wesley Ivan Hurt, Nov 01 2013 *)

a(n) = 3 * 2^(n\2 + 1 + (-1)^n); \\ Michel Marcus, Nov 02 2013

def A112033(n): return 3*(1<<(n>>1)+(int(not n&1)<<1)) # Chai Wah Wu, Jan 17 2023

a(n) = 1 / abs(A112031(n)/A112032(n) - 2/3). (previous name)
a(n) = 3*2^A084964(n) = 3*A112032(n).
From Ralf Stephan, Jul 16 2013: (Start)
Recurrence: a(n) = 2a(n-2), a(0)=12, a(1)=3.
G.f.: (6*x+24)/(1-2*x^2). (End)
From Amiram Eldar, May 11 2025: (Start)
Sum_{n>=0} 1/a(n) = 5/6.
Sum_{n>=0} (-1)^n/a(n) = -1/2. (End)

A112034 1 / (A010684(n)/A016116(n+5) - 1/A112033(n)).

Original entry on oeis.org

6, 24, 12, 48, 24, 96, 48, 192, 96, 384, 192, 768, 384, 1536, 768, 3072, 1536, 6144, 3072, 12288, 6144, 24576, 12288, 49152, 24576, 98304, 49152, 196608, 98304, 393216, 19660, 8, 786432, 393216, 1572864, 786432, 3145728, 1572864, 6291456

Offset: 0

Reinhard Zumkeller, Aug 27 2005

nonn

a(n) = 3*2^A052938(n).

Cf. A112030, A112031, A112032.

cp's OEIS Frontend

A112030 a(n) = (2 + (-1)^n) * (-1)^floor(n/2).

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A112032 Denominator of 3/4 + 1/4 - 3/8 - 1/8 + 3/16 + 1/16 - 3/32 - 1/32 + 3/64 ...

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A112033 a(n) = 3 * 2^(floor(n/2) + 1 + (-1)^n).

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A112034 1 / (A010684(n)/A016116(n+5) - 1/A112033(n)).

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